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A363452 Total number of blocks containing only odd elements in all partitions of [n].

%I A363452 #19 Dec 08 2023 07:12:15
%S A363452 0,1,1,5,12,62,206,1189,4949,31775,156972,1110280,6301550,48637701,
%T A363452 310279615,2591820857,18293310174,164218811718,1267153412532,
%U A363452 12152174863961,101557600812015,1035203191874931,9299499328238110,100314319611860936,962663031508255416
%N A363452 Total number of blocks containing only odd elements in all partitions of [n].
%H A363452 Alois P. Heinz, <a href="/A363452/b363452.txt">Table of n, a(n) for n = 0..250</a>
%H A363452 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363452 a(n) = Sum_{k=0..ceiling(n/2)} k * A124420(n,k).
%F A363452 a(n) = A363434(n) - A363453(n).
%F A363452 a(2n) = A363453(2n).
%F A363452 a(2n+1) = A363453(2n+1) + A094577(n).
%e A363452 a(3) = 5 = 0 + 1 + 1 + 1 + 2 : 123, 12|3, 13|2, 1|23, 1|2|3.
%p A363452 b:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
%p A363452       add(Stirling2(i, k)*binomial(u, i)*
%p A363452       add(Stirling2(g, j)*j^(u-i), j=0..g), i=k..u)
%p A363452     end:
%p A363452 a:= n-> add(b(n, k)*k, k=0..ceil(n/2)):
%p A363452 seq(a(n), n=0..25);
%p A363452 # second Maple program:
%p A363452 b:= proc(n, x, y, m) option remember; `if`(n=0, y,
%p A363452       `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
%p A363452       `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
%p A363452     end:
%p A363452 a:= n-> b(n, 0$3):
%p A363452 seq(a(n), n=0..25);
%t A363452 b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, y,
%t A363452      If[x + m > 0, b[n-1, y, x, m]*(x+m), 0] + b[n-1, y, x+1, m] +
%t A363452      If[y > 0, b[n-1, y-1, x, m+1]*y, 0]];
%t A363452 a[n_] := b[n, 0, 0, 0];
%t A363452 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 08 2023, after _Alois P. Heinz_ *)
%Y A363452 Cf. A000110, A094577, A124420, A363434, A363453.
%K A363452 nonn
%O A363452 0,4
%A A363452 _Alois P. Heinz_, Jun 02 2023